# Intro/Motivation

## Space, but Which One?

• You run into a “space” in the wild. Which one is it?
• How many possible spaces could it be?
• How much information is needed to specify our space uniquely?

## Another Application: Data

• Possible to fit data to a high-dimensional manifold
• Makes clustering/grouping easier
• (Here, slice with a hyperplane)
• Extract info about an entire family of objects and how they vary.
• Also useful for outlier detection!

## Where to Start

• What structure does your space have?

• What can you “measure” locally?

• How might it vary in ways you can’t measure?

• Important question before attempting to classify:

1. What does “space” mean?
• Need to pick a category to work in.
1. What does “which” mean?
• How to distinguish? Need an equivalence relation!

# Q1: Types of Spaces

• Plan: compare classification theorem in topology and algebraic geometry
• Hopefully see some fun spaces along the way!
• But first: address classification and the notion of “sameness”

## The Greeks: Conics

• Early classification efforts: conic sections.

• Apollonius, 190 BC, Ancient Greeks
• Key idea: realize as intersection loci in bigger space

• (Projectivize $\mathbb{R}^2$.)

• A conic is specified by 6 parameters:

\begin{align*} A x^{2}+B x y+C y^{2}+D x+E y+F=0 \\ \\ Q :=\left(\begin{array}{ccc} A & B / 2 & D / 2 \\ B / 2 & C & E / 2 \\ D / 2 & E / 2 & F \end{array}\right), \quad \mathbf{x} = [x, y, 1] \\ \implies \mathbf{x}^t Q \mathbf{x} = 0 \end{align*}

$\det(Q)$ Conic
$<0$ Hyperbola
$=0$ Parabola
$>0$ Ellipse
• Each conic is a variety

• Can obtain every conic by "modulating* 6 parameters.
• $\mathbb{R}^6$: too much information: scaling by a nonzero $\lambda \in \mathbb{R}$ yields the same conic, so reduce the space $[A, B, C, D, E, F]\in \mathbb{R}^6 \mapsto [A: B: C: D: E: F] \in \mathbb{RP}^5 .$

• Important point: $\mathbb{RP}^5$ is a projective variety and a smooth manifold! Tools available:

• Dimension (what does a generic point look like?)
• Tangent and cotangent spaces, differential forms
• Measures, metrics, volumes, integrals
• Intersection theory (Bezout’s Theorem!), subvarieties, curves
• Linear algebra and Combinatorics (enumerative questions)
• We can imagine a moduli space of conics that parameterizes these:

Some Calc III review:

General form: $\begin{array}{l}\scriptsize A x^{2}+B y^{2}+C z^{2}+2 F y z+2 G z x+2 H x y+2 P x+2 Q y+2 R z +D=0 \\ \\ \text{Setting}\,\, E:= \left[\begin{array}{llll} A & H & Q & P \\ H & B & F & Q \\ G & F & C & R \\ P & Q & R & D \end{array}\right]\,\, e:= \left[\begin{array}{lll} A & H & G \\ H & B & F \\ G & F & C \end{array}\right] \\ \Delta := \operatorname{det}(E) \end{array}$

(discriminants), the equation becomes $\mathbf x^t E \mathbf x = 0$ and we have a classification:

What is the moduli space? It sits inside $\mathbb{R}^{10}$, possibly $\mathbb{RP}^{9}$ but not in the literature!

## Automorphisms

• Problem: infinitely many points in these moduli spaces correspond to the same “class” of conic
• How to address: Klein’s Erlangen program, understand the geometry of a space by understanding its structure-preserving automorphisms.

• For topological spaces: a Lie group acting on the space.

• Can then “mod out” by the appropriate morphisms to (hopefully) get finitely many equivalence classes

# What Does “Space” Mean?

## Some Setup

• Algebraic Variety: Irreducible ,zero locus of some family $f\in \mathbb{k}[x_1, \cdots, x_n]$ in $\mathbb{A}^n/\mathbb{k}$.
• Equivalently, a locally ringed space $(X, \mathcal{O}_X)$ where $\mathcal{O}_X$ is a sheaf of finite rational maps to $\mathbb{k}$.
• Projective Variety: Irreducible zero locus of some family $f_n \subset \mathbb{k}[x_0, \cdots, x_n]$ in $\mathbb{P}^n/\mathbb{k}$
• Admits an embedding into $\mathbb{P}^\infty/\mathbb{k}$ as a closed subvariety.
• Dimension of a variety: the $n$ appearing above.
• Topological Manifold: Hausdorff, 2nd Countable, topological space, locally homeomorphic to $\mathbb{R}^n$.
• Equivalently, a locally ringed space where $\mathcal{O}_X$ is a sheaf of continuous maps to $\mathbb{R}^n$.
• Smooth Manifold: Topological manifold with a smooth structure (maximal smooth atlas) with $C^\infty$ transition functions.
• Equivalently, a locally ringed space where $\mathcal{O}_X$ is a sheaf of smooth maps to $\mathbb{R}^n$.
• Algebraic Manifold: A manifold that is also a variety, i.e. cut out by polynomial equations. Example: $S^n$.
• Manifolds will be compact and without boundary, varieties are (probably) smooth, separated, of finite type.

## Impossible Goal

• Pick a category, understand all of the objects (identifying a moduli “space”) and all of the maps.
• Understand all topological spaces up to ???
• Homeomorphism?
• Diffeomorphism?
• Homotopy-Equivalence?
• Cobordism?
• Understand all algebraic and/or projective varieties up to
• Biregular maps?
• Birational maps?
• Locally ringed morphisms?

# Classification in Topology

## Topological Category

• Classifying manifolds up to homeomorphism: stratify “moduli space” of topological manifolds by dimension.

• Dimensions 0,1,2,3:

• Smooth = Top. See smooth classification.
• Dimension 4:
• Topologically classified by surgery, but barely, and not smoothly.
• Dimension $n\geq 5$:
• Uniformly “classified” by surgery, s-cobordism, with a caveat:
• $\pi_1$ can be any finitely presented group – word problem
• Instead, breaks homotopy type of a fixed manifold up into homeomorphism classes

## Smooth Category

Generally expect things to split into more classes.

• Dimension 0: The point (terminal object)
• Dimension 1: $\mathbb{S}^1, \mathbb{R}^1$
• Dimension 2: $\left\langle\mathbb{S}^2, \mathbb{T}^2, \mathbb{RP}^2 \mid \mathbb{S}^2 = 0,\,\,3\mathbb{RP}^2 = \mathbb{RP}^2 + \mathbb{T}^2 \right\rangle$.
• Classified by $\pi_1$ (orientability and “genus”). Riemann, Poincaré, Klein.

• Dimension 2: closed + orientable $\implies$ complex
• Uniformization: Holomorphically equivalent to a quotient of one of three spaces/geometries:
• $\mathbb{CP}^1$, positive curvature (spherical)
• $\mathbb{C}$, zero curvature (flat, Euclidean)
• $\mathbb{H}$ (equiv. $\mathbb{D}^\circ$), negative curvature (hyperbolic)
• Stratified by genus
• Genus 0: Only $\mathbb{CP}^1$
• Genus 1: All of the form $\mathbb{C}/\Lambda$, with a distinguished point $[0]$, i.e. an elliptic curve.
• Has a topological group structure!
• Genus $\geq 2$: Complicated?

Doesn’t capture holomorphy type completely.

## 3-manifolds: Thurston’s Geometrization

• Geometric structure: a diffeo $M\cong \tilde M/\Gamma$ where $\Gamma$ is a discrete Lie group acting freely/transitively on $X$ (as in Erlangen program)
• Decompose into pieces with one of 8 geometries:
• Spherical $\sim S^3$
• Euclidean $\sim \mathbb{R}^3$
• Hyperbolic $\sim \mathbb{H}^3$
• $S^2\times \mathbb{R}$
• $\mathbb{H}^2\times \mathbb{R}$
• $\widetilde{\mathrm{SL}(2, \mathbb{R})}$
• “Nil”
• “Sol”
• Proved by Perelman 2003, Ricci flow with surgery.
• 4-manifolds: classified in the topological category by surgery, but not in the smooth category
• Hard! Will examine special cases of Calabi-Yau
• Open part of Poincaré Conjecture.
• Dimension $\geq 5$: surgery theory, strong relation between diffeomorphic and s-cobordant.

## Toward Algebraic Manifolds: Berger’s Classification

• Every smooth manifold admits a Riemannian metric, so consider Riemannian manifolds

• Here $H\leq \mathrm{SO}(n)$ is the holonomy group:

• Berger’s classification for smooth Riemannian manifolds, one of 7 possibilities. $\tiny \begin{array}{|c|c|c|c|c|} \hline n=\operatorname{dim} M & H & \text { Parallel tensors } & \text { Name } & \text { Curvature } \\ \hline n & \mathrm{SO}(n) & g, \mu & \text {orientable} & \\ \hline 2 m(m \geq 2) & \mathrm{U}(m) & g, \omega & \textbf{Kähler} & \\ \hline 2 m(m \geq 2) & \mathrm{SU}(m) & g, \omega, \Omega & \textbf{Calabi-Yau} & \text {Ricci-flat} \\ \hline 4 m(m \geq 2) & \mathrm{Sp}(m) & g, \omega_{1}, \omega_{2}, \omega_{3}, J_{1}, J_{2}, J_{3} & \textbf{hyper-Kähler} & \text {Ricci-flat} \\ \hline 4 m(m \geq 2) & (\mathrm{Sp}(m) \times \mathrm{Sp}(1)) / \mathbb{Z}_{2} & g, \Upsilon & \text {quaternionic-Kähler} & \text {Einstein} \\ \hline 7 & \mathrm{G}_{2} & g, \varphi, \psi & \mathrm{G}_{2} & \text {Ricci-flat} \\ \hline 8 & \operatorname{Spin}(7) & g, \Phi & \operatorname{Spin}(7) & \text {Ricci-flat} \\ \hline \end{array}$

Types in bold: amenable to Algebraic Geometry. $G2$ shows up in Physics!

• Ricci-flat, i.e. Ricci curvature tensor vanishes
• (Measures deviation of volumes of “geodesic balls” from Euclidean balls of the same radius)

# Classification in Algebraic Geometry

## Enriques-Kodaira Classification

Work over $\mathbb{C}$ for simplicity, take all dimensions over $\mathbb{C}$.

• Minimal model program: classifying complex projective varieties.

• Stratify the “moduli space” of varieties by $\mathbb{k}-$dimension.

• Dimension 1:

• Smooth Algebraic curves = compact Riemann surfaces, classified by genus
• Roughly known by Riemann: moduli space of smooth projective curves $\mathcal{M}_g$ is a connected open subset of a projective variety of dimension $3g-3$.

We’ll come back to these!

• Dimension 2:
• Smooth Algebraic Surfaces: Hard. See Enriques classification.

• Setting of classical theorem: always 27 lines on a cubic surface!

• Example Clebsch surface, satisfies the system $\begin{array}{l} x_{0}+x_{1}+x_{2}+x_{3}+x_{4}=0 \\ \\ x_{0}^{3}+x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}=0 \end{array}$

## Interesting Space: Elliptic Curves

• Equivalently, Riemann surfaces with one marked point.
• Equivalently, $\mathbb{C}/\Lambda$ a lattice, where homothetic lattices (multiplication by $\lambda \in \mathbb{C}- \{0\}$) are equivalent.
• Generalize to $\mathbb{C}^n/\Lambda$ to obtain abelian varieties.

## Interesting Space: Moduli of Elliptic Curves

• $\mathcal{M}_g$: the moduli space of compact Riemann surfaces (curves) of genus $g$, i.e. elliptic curves.

## Dimension 2: Algebraic Surfaces

• Definition: Kodaira Dimension

• $X$ has some canonical sheaf $\omega_X$, you can take some sheaf cohomology and get a sequence of integers (plurigenera)

\begin{align*} P_{\mathbf{n}} (X) &:= h^0(X, \omega_X^{\otimes \mathbf{n}}) \quad n\in \mathbb{Z}^{\geq 0} \\ \\ \implies \kappa(X) &:= \limsup_{\mathbf n \to \infty} {\log P_{\mathbf n}(X) \over \log(\mathbf{n}) } \end{align*}

## Dimension 2: Algebraic Surfaces

Every such surface has a minimal model of one of 10 types:

$\kappa = -\infty$ (2 main types)

1. Rational: $\cong \mathbb{CP}^2$
1. Ruled: $\cong X$ for $\mathbb{CP}^1 \to X \to C$ a bundle over a curve.
• Called “ruled” because every point is on some $\mathbb{CP}^1$.
1. Type VII

$\kappa = 0$ (Elliptic-ish, 4 types)

1. Enriques (all (quasi)-elliptic fibrations)
1. Hyperelliptic
• Taking Albanese embedding (generalizes Jacobian for curves) yields an elliptic fibration
• (i.e. a surface bundle, potentially with singular fibers)
1. $K3$ (Kummer-Kahler-Kodaira) surfaces
1. Toric and Abelian Surfaces:
• 2 dimensional abelian varieties (projective algebraic variety + algebraic group structure).
• Compare to 1 dimensional case: all 1d complex torii are algebraic varieties,
• Riemann discovered that most 2d torii are not.
1. Kodaira Surfaces

$\kappa = 1$: Other elliptic surfaces

1. Properly quasi-elliptic.

Elliptic fibration, but almost all fibers have a node.

$\kappa = 2$ (Max possible, “everything else”)

1. General type

# Interesting Space: Toric Varieties

• Definitions:
• Define a complex torus as $\mathbb{T} = (\mathbb{C}^{\times})^n \subseteq \mathbb{C}^n$

• Can be written as the zero set of some $f\in \mathbb{C}[x_0, \cdots, x_n]$ in $\mathbb{C}^{n+1}$.

Generalizes to algebraic groups over a field: $(\mathbb{G}_m)^n$ (analogy: maximal torus/Cartan subalgebra in Lie theory)

• Toric variety: $X$ contains a dense Zariski-open torus $\mathbb{T}$, where the action of $\mathbb{T}$ on itself as a group extends to $X$.

• Flavor: spaces modeled on convex polyhedra
• Examples: bundles over $\mathbb{CP}^n$.
• Why study:
• Model spaces by rigid geometry, generalize things like Bezier curves
• Some are determined by rigid combinatorial data (“fan”, or polytopes)
• Combinatorial data for constructions in mirror symmetry, e.g. Calabi-Yaus (1/2 of one billion threefolds!)

## Kahler Manifolds/Varieties

• As complex manifolds:
• A symplectic manifold $(X, \omega)$ with an integrable almost-complex structure $J$ compatible with $\omega$.
• Yields an inner product on tangent vectors: $g(u, v) := \omega(u, Jv)$ (i.e. a metric)

• Examples: all smooth complex projective varieties
• But not all complex manifolds (exception: Stein manifolds)
• Specialize to Calabi-Yaus: compact, Ricci-flat, trivial canonical.
• Calabi’s Conjecture and Yau’s field medal: existence of Ricci-flat Kahlers (Calabi-Yaus)

Trivial canonical $\implies$ exists a nowhere vanishing top form = top wedge of $T^* X$ is the trivial line bundle

## Calabi-Yau

• Another from Berger’s classification, special case of Kahler
• Applications: Physicists want to study $G_2$ manifolds (an exceptional Lie group, automorphisms of octonions)
• Part of $M$-theory uniting several superstring theories, but no smooth or complex structures.
• Indirect approach: compactify an 11-dimension space, one small $S^1$ dimension $\to$ 10 dimensions
• 4 spacetime and 6 “small” Calabi-Yau
• Superstring theory: a bundle over spacetime with fibers equal to Calabi-Yaus.

Roughly, genera of fibers will correspond to families of observed particles.

## Calabi-Yaus

• As manifolds:
• Ricci-flat: vacuum solutions to (analogs of) Einstein’s equations with zero cosmological constant

• Setting for mirror symmetry: the symplectic geometry of a Calabi-Yau is “the same” as the complex geometry of its mirror.

• Yau, Fields Medal 1982: There are Ricci flat but non-flat (nontrivial holonomy) projective complex manifolds of dimensions $\geq 2$.

• As varieties: the canonical bundle $\Lambda^n T^* V$ is trivial

• Compact classification for $\mathbb{C}-$dimension:

• Dimension 1: 1 type, all elliptic curves (up to homeomorphism)
• Dimension 2: 1 type, $K3$ surfaces
• Dimension 3: (threefolds) conjectured to be a bounded number, but unknown.
• At least 473,800,776!